Bounded Proof : If {a_n + b_n} and {a_n} are both bounded, then {b_n} is bounded.

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• Proof Equivalent Cauchy Sequences

  Show that if (a_n)^infinity evaluated at n=1 and (b_n)^infinity evaluated at n=1 are eventually epsilon-close, then (a_n)^infinity evaluated at n=1 is bounded if and only if (b_n)^infinity evaluated at n=1 is bounded. ** Please see the attached file for

• Proof : Sequences and Supremum

  70592 Proof: Sequences and Supremum Suppose that the sequences {a_n}_n is bounded above and lim(b_n) exists. a) Prove that for all e>0 there is an N st that for all n>=N sup{a_k:k>=n} + b_n <=sup{a_k + b_k: k>=n} + e.

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• Equivalent sequnces: boundness and Cauchy

  Show that if (a_n)[n=1,infinity] and (b_n)[n=1,infinity] are eventually epsilon-close, then (a_n)[n=1,infinity] is bounded if and only if (b_n)[n=1,infinity] is bounded. Please refer to the attachment for clarification, in case needed.

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• Proofs : Bounded Sequences

  70591 Proofs : Bounded Sequences Suppose the sequences {a_n}_n and {b_n}_n are both bounded above.

• Difference Quotient : Limits and Differentiable Functions

  153549 Difference Quotient : Limits and Differentiable Functions Assume f:(-1,1) --> R and f'(0) exists. If a_n , b_n -> 0 as n->infinity, define the difference quotient: D_n = ( f(b_n) - f(a_n) ) / ( b_n - a_n).

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